Question

Help!! it says only numeric answers allowed

Answer 1

Q4.The focal point of the parabola
`$y = x^2 + 1$` is
`$(0, 1 + 1) = (0, 2)$`.

So the answer is p = 2.

Q5.The value of l is 1.

Q4.The focal point of a parabola is a point that is equidistant to the parabola and its directrix. The directrix is a line that is parallel to the axis of symmetry of the parabola.

In the case of the parabola
`$y = x^2 + 1$`, the axis of symmetry is the y-axis. The directrix is the line y = -1.

To find the distance between a point (x, y) on the parabola and the directrix, we need to find the equation of the line that is perpendicular to the directrix and passes through (x, y). The slope of the directrix is $0$, so the slope of the perpendicular line will be undefined. This means that the perpendicular line will be a vertical line.

The equation of the vertical line that passes through
`$(x, y)$` is
`$x = x$.`

The distance between the point
`$(x, y)$` and the line
`$y = -1$` is the length of the segment that is perpendicular to both lines. This segment is called the focal length.

The focal length is equal to the distance between the vertex of the parabola and the focal point. In the case of the parabola
`$y = x^2 + 1$`, the vertex is at (0, 1), so the focal length is 1.

Therefore, the focal point of the parabola
`$y = x^2 + 1$` is
`$(0, 1 + 1) = (0, 2)$`.

So the answer is p = 2.

Q5.Sure, I can help. The image you sent me shows a parabola with a directrix labeled as y = 1. The question asks for the value of l.

The directrix of a parabola is a line that is equidistant to the parabola and its focus. In the image, the directrix is horizontal, which means the parabola is symmetric about its vertical axis.

To find the value of l, we need to know the distance between the vertex of the parabola and the directrix. We can see from the image that the vertex is halfway between the directrix and the topmost point of the parabola. Since the distance between the directrix and the topmost point of the parabola is 2 units, the distance between the vertex and the directrix is 1 unit.